52 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
A general approach to transversal versions of Dirac-type theorems
Given a collection of hypergraphs with the same
vertex set, an -edge graph is a transversal if
there is a bijection such that for
each . How large does the minimum degree of each need to be so
that necessarily contains a copy of that is a transversal?
Each in the collection could be the same hypergraph, hence the minimum
degree of each needs to be large enough to ensure that .
Since its general introduction by Joos and Kim [Bull. Lond. Math. Soc., 2020,
52(3):498-504], a growing body of work has shown that in many cases this lower
bound is tight. In this paper, we give a unified approach to this problem by
providing a widely applicable sufficient condition for this lower bound to be
asymptotically tight. This is general enough to recover many previous results
in the area and obtain novel transversal variants of several classical
Dirac-type results for (powers of) Hamilton cycles. For example, we derive that
any collection of graphs on an -vertex set, each with minimum degree at
least , contains a transversal copy of the -th power of a
Hamilton cycle. This can be viewed as a rainbow version of the P\'osa-Seymour
conjecture.Comment: 21 pages, 4 figures; final version as accepted for publication in the
Bulletin of the London Mathematical Societ
The hitting time of clique factors
In a recent paper, Kahn gave the strongest possible, affirmative, answer to
Shamir's problem, which had been open since the late 1970s: Let and
let be divisible by . Then, in the random -uniform hypergraph process
on vertices, as soon as the last isolated vertex disappears, a perfect
matching emerges. In the present work, we transfer this hitting time result to
the setting of clique factors in the random graph process: At the time that the
last vertex joins a copy of the complete graph , the random graph process
contains a -factor. Our proof draws on a novel sequence of couplings,
extending techniques of Riordan and the first author. An analogous result is
proved for clique factors in the -uniform hypergraph process ()
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Paths and cycles in graphs and hypergraphs
In this thesis we present new results in graph and hypergraph theory all of which feature paths or cycles.
A -uniform tight cycle is a -uniform hypergraph on vertices with a cyclic ordering of its vertices such that the edges are all -sets of consecutive vertices in the ordering.
We consider a generalisation of Lehel's Conjecture, which states that every 2-edge-coloured complete graph can be partitioned into two cycles of distinct colour, to -uniform hypergraphs and prove results in the 4- and 5-uniform case.
For a -uniform hypergraph~, the Ramsey number is the smallest integer such that any 2-edge-colouring of the complete -uniform hypergraph on vertices contains a monochromatic copy of . We determine the Ramsey number for 4-uniform tight cycles asymptotically in the case where the length of the cycle is divisible by 4, by showing that = (5+(1)).
We prove a resilience result for tight Hamiltonicity in random hypergraphs. More precisely, we show that for any >0 and 3 asymptotically almost surely, every subgraph of the binomial random -uniform hypergraph in which all -sets are contained in at least edges has a tight Hamilton cycle.
A random graph model on a host graph is said to be 1-independent if for every pair of vertex-disjoint subsets of , the state of edges (absent or present) in is independent of the state of edges in . We show that = 4 - 2 is the critical probability such that every 1-independent graph model on where each edge is present with probability at least contains an infinite path
A general approach to transversal versions of Dirac-type theorems
Given a collection of hypergraphs =(1,...,) with the same vertex set, an -edge graph ⊂∪∈[] is atransversal if there is a bijection ∶()→[] such that ∈(()) for each ∈(). How large does the minimum degree of each need to be so that necessarily contains a copy of that is a transversal? Each in the collection could be the same hypergraph,hence the minimum degree of each needs to be large enough to ensure that ⊆. Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020)498–504), a growing body of work has shown that inmany cases this lower bound is tight. In this paper, wegive a unified approach to this problem by providinga widely applicable sufficient condition for this lowerbound to be asymptotically tight. This is general enoughto recover many previous results in the area and obtainnovel transversal variants of several classical Dirac-typeresults for (powers of) Hamilton cycles. For example, wederive that any collection of graphs on an -vertex set, each with minimum degree at least (∕( + 1) +(1)), contains a transversal copy of the th power of a Hamilton cycle. This can be viewed as a rainbow versionof the Pósa–Seymour conjecture
A general approach to transversal versions of Dirac-type theorems
Given a collection of hypergraphs (Formula presented.) with the same vertex set, an (Formula presented.) -edge graph (Formula presented.) is a transversal if there is a bijection (Formula presented.) such that (Formula presented.) for each (Formula presented.). How large does the minimum degree of each (Formula presented.) need to be so that (Formula presented.) necessarily contains a copy of (Formula presented.) that is a transversal? Each (Formula presented.) in the collection could be the same hypergraph, hence the minimum degree of each (Formula presented.) needs to be large enough to ensure that (Formula presented.). Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020) 498–504), a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles. For example, we derive that any collection of (Formula presented.) graphs on an (Formula presented.) -vertex set, each with minimum degree at least (Formula presented.), contains a transversal copy of the (Formula presented.) th power of a Hamilton cycle. This can be viewed as a rainbow version of the Pósa–Seymour conjecture
Resilience for Loose Hamilton Cycles
We study the emergence of loose Hamilton cycles in subgraphs of random
hypergraphs. Our main result states that the minimum -degree threshold for
loose Hamiltonicity relative to the random -uniform hypergraph
coincides with its dense analogue whenever . The
value of is approximately tight for . This is particularly
interesting because the dense threshold itself is not known beyond the cases
when .Comment: 33 pages, 3 figure
Perfect matchings in random sparsifications of Dirac hypergraphs
For all integers , let be the minimum
integer such that every -uniform -vertex hypergraph with minimum -degree at least has an optimal
matching. For every fixed integer , we show that for and , if is an -vertex
-uniform hypergraph with , then
a.a.s.\ its -random subhypergraph contains a perfect matching
( was determined by R\"{o}dl, Ruci\'nski, and Szemer\'edi for all
large ). Moreover, for every fixed integer and
, we show that the same conclusion holds if is an
-vertex -uniform hypergraph with . Both of these results strengthen Johansson, Kahn,
and Vu's seminal solution to Shamir's problem and can be viewed as "robust"
versions of hypergraph Dirac-type results. In addition, we also show that in
both cases above, has at least many perfect matchings, which is best possible up to a
factor.Comment: 25 pages + 2 page appendix; Theorem 1.5 was proved in independent
work of Pham, Sah, Sawhney, and Simkin (arxiv:2210.03064
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